What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a real valued which directly gives the probability. My question is why should we use complex valued state functions? There must be some strong non-mathematical reason which I am not able to find in books I have referred?

The commutator $[x,p]=i\hbar$ contains an $i$ - and it must contain an $i$ because the commutator of two Hermitian operators is anti-Hermitian - so it can't be represented by real operators on a real vector space. Equivalently, $\cos(kx)$ instead of $\exp(ikx)$ wouldn't remember whether one is moving left or right, or the sign of energy etc. One couldn't write first-order oscillating equations without $i$. See also Why complex numbers are fundamental in physics, motls.blogspot.com/2010/08/…
–
Luboš MotlNov 18 '11 at 8:54

Thanks @yayu : But here my question is in a bit naive sense (i have just begun reading QM). Is there any physical significance of the real and imaginary parts of $\Psi$ and the phase of $\Psi$. Why don't we give them some names which could give some intuitive appeal ?
–
Rajesh DNov 18 '11 at 12:44

On a related note, Gary Gibbons has written a paper discussing the fundamentality of complex numbers in quantum mechanics.
–
twistor59Nov 18 '11 at 16:28

3 Answers
3

One of the most important aspects of QM is interference. Think of the double-slit experiment: you want the wave functions to add with their relative phases, rather than add the probabilities.

For this, wave functions must have a phase, and complex numbers capture this naturally.

The absolute phase of the wave function is not measurable, and contains no physical information. The only thing that matters is realtive phase. Therefore, the real or imaginary parts are also useless on their own.

Feynman explains this nicely in "lectures on physics" (volume III if I recall correctly)

The reason is that there is interference in quantum mechanics. Forget the complex values, after decomposing a complex number into real and imaginary parts, the complex nature of $\psi$ is just a doubling of all states (each state has a real and imaginary verson), and this doubling is interesting and important, but less essential than the signs.

The main point is that the quantum mechanical calculus involves the same rules as probability for consecutive and independent events, you multiply the real-valued amplitudes for consecutive events, and add them up for independent events (when you duplicate all states, there are always alternate paths which separately give the real and imaginary contributions, and this is best given by using complex numbers, but this is a separate issue). But you can have cancellations between different possibilities, destructive interferences, so that the quantum amplitudes contain signed probability-like objects, and these are called probability amplitudes.

The essential reason that $\psi$ is not a probability is that the quantities which describe physical likelihoods are the signed probability amplitudes, which allow for interference effects.

To see why you need a doubling of states and the associated complex structure, you should consider particles propagating in the forward and backward direction. The only difference between the directions is the sense of the phase variation along the complex wave. The time-reversing operator gives the complex conjugation relation, and this requires the doubling of states into complex numbers. The need for a complex state space (or an equivalent real construction) is apparent in the canonical commutation relations, as Lubos Motl points out in the comments.

I would say the main reason why we should use complex valued state functions is it may be more convenient this way. However, it is not obvious though that we must use complex valued functions. Source: http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (an article published in the Journal of Mathematical Physics).
Briefly: in a general case, three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation, yielding an equivalent fourth-order partial differential equation. The remaining component can be made real (at least locally) by a gauge transform. Thus, the Dirac equation can be replaced by an equation for just one real function. Let me emphasize that complex numbers are not just replaced by pairs of real numbers in this way. See also the reference there to a predecessor - an old Schrödinger's article, where a similar procedure was suggested for a scalar field.